Differential Forms and Electromagnetic Field Theory

Transcription

1 Progress In Eletromagnetis Researh, Vol. 148, , 2014 Differential Forms and Eletromagneti Field Theory Karl F. Warnik 1, * and Peter Russer 2 (Invited Paper) Abstrat Mathematial frameworks for representing fields and waves and expressing Maxwell s equations of eletromagnetism inlude vetor alulus, differential forms, dyadis, bivetors, tensors, quaternions, and Clifford algebras. Vetor notation is by far the most widely used, partiularly in appliations. Of the more sophistiated notations, differential forms stand out as being lose enough to vetors that most pratitioners an readily understand the notation, yet at the same time offering unique visualization tools and graphial insight into the behavior of fields and waves. We survey reent papers and book on differential forms and review the basi onepts, notation, graphial representations, and key appliations of the differential forms notation to Maxwell s equations and eletromagneti field theory. 1. INTRODUCTION Sine James Clerk Maxwell s disovery of the full set of mathematial laws that govern eletromagneti fields, other mathematiians, physiists, and engineers have proposed a surprisingly large number of mathematial frameworks for representing fields and waves and working with eletromagneti theory. Vetor notation is by far the most ommon in textbooks and engineering work, and has the advantage of being widely taught and learned. More advaned analytial tools have been developed, inluding dyadis, bivetors, tensors, quaternions, and Clifford algebras. The ommon theme of all of these mathematial notations is to hide the omplexity of the set of 20 oupled differential equations in Maxwell s original papers [1, 2] using high level abstrations for field and soure quantities. Inreasing the sophistiation of the notation simplifies the appearane of the governing equations, revealing hidden symmetries and deeper meaning in the equations of eletromagnetism. For various reasons, however, few of the available higher order mathematial notations are atually used by applied pratitioners and engineers. Tensors, Clifford algebras, and other related analysis tools are not widely taught, partiularly in engineering programs. The onundrum is that regardless of the elegane of the formulation, to solve onrete problems the omplexity of the mathematial symbols must be unwound to obtain formulas involving omponents of the eletromagneti field in a speifi oordinate system. There is something of a balaning at when hoosing a mathematial formalism if the notation is too abstrat, it is hard to solve pratial problems, but at the same time a high-level representation an greatly aid in learning and understanding the underlying priniples of eletromagneti theory. For day-to-day work in eletromagnetis engineering, Heaviside s vetor analysis [3] is a near-perfet ompromise between abstration and onreteness, and when lightly flavored with dyadi notation, adequate for nearly all pratial problems. Bearing in mind this need for balane between onreteness and abstration, among the nontraditional notations for eletromagneti theory there is one that stands out: the alulus of differential Reeived 30 June 2014, Aepted 19 July 2014, Sheduled 21 July 2014 Invited paper for the Commemorative Colletion on the 150-Year Anniversary of Maxwell s Equations. * Corresponding author: Karl F. Warnik (warnik@ee.byu.edu). 1 Department of Eletrial and Computer Engineering, Brigham Young University, 459 Clyde Building, Provo, UT 84602, USA. 2 Institute for Nanoeletronis, Tehnial University Munih, Arisstrasse 21, Munih 80333, Germany.

2 84 Warnik and Russer forms. Differential forms have the advantage of behaving muh like vetors, making the notation far easier to learn than, say, tensor analysis. At the same time, differential forms notation simplifies some alulations and allows for elegant pitures that larify some of the non-intuitive aspets of eletromagneti fields. This makes differential forms an attrative supplement to vetor notation, and worth spending some time with to enhane one s deeper understanding of eletromagneti theory. Differential forms originated in the work of Hermann Günter Grassmann and Élie Cartan. In 1844, Hermann Günter Grassmann published his book Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik [4], in whih he developed the idea of an algebra in whih the symbols representing geometri entities suh as points, lines, and planes are manipulated using ertain rules. Grassmann introdued what is now alled exterior algebra, based upon the exterior produt a b = b a (1) Based on Grassmann s exterior algebra Cartan [5] developed the exterior alulus. Exterior alulus has proven to be the natural language of field theory sine it yields simple and ompat mathematial formulae, and simplifies the solution of field theoretial problems. Differential forms establish a diret onnetion to geometrial images and provide additional physial insight into eletromagnetism. Eletromagneti theory merges physial, mathematial, and geometrial ideas. In this omplex environment reative thinking and the onstrution of new onepts is supported by imagery as provided by geometri models [6]. Sine the introdution of the differential forms notation or exterior alulus, the onnetion between the mathematial notation and geometri interpretations of field and soure quantities has been well established in the literature on eletromagneti field theory. Early books that popularized the use of differential forms for eletromagnetism inlude the oldshool texts by Flanders [7] and the surprisingly fun general relativity textbook by Misner et al. [8], whih ombined rigorous mathematis with lever and interesting graphial illustrations. Another early text that adopted the notation was Thirring [9]. The well-known paper by Deshamps [10] did muh to entie students of eletromagnetism to this notation. Later ame the ounterulture text by Burke [11]. Early papers on various aspets of differential forms and eletromagnetis inlude [12 19]. Sine those pioneering works, a number of intermediate and advaned eletromagnetis textbooks using differential forms have been published [20 22]. Mathematially oriented books inlude [23, 24]. One of the authors has published an introdution into basi eletromagnetis priniples and their appliations to antenna and mirowave iruit design based on exterior differential alulus [25, 26], and both authors have added the publiation of a problem solving book presenting a wide range of real-world eletromagnetis problems [27]. Hehl and Obukhov have published an introdution into eletrodynamis based on exterior alulus [28], and Lindell has published a omprehensive treatment of differential forms in eletromagnetis [29]. The breadth of the topi has been expanded by treatments of boundary onditions [30], eletromagnetis in two and four dimensions [31], appliations of Green s theorem [32], Green s funtions [33, 34], and onstitutive relations [35]. The use of differential forms in undergraduate teahing was promoted in [36]. Other relevant papers over the last two deades inlude [37, 38]. Numerial methods based on finite elements and other mathematial frameworks have been heavily impated by differential forms [39 41]. In appliations to numerial analysis, disrete differential forms is an area of partiular noteworthiness [42 44]. Absorbing boundary onditions were treated by Teixeira and Chew [45]. Transformation optis also has a lose relationship to differential forms. Theoretial treatments prefigured these results, inluding [46]. A formula that related the Hodge star operator in non-eulidean geometry to the permittivity and permeability of an inhomogeneous, anisotropi medium first appeared in [47]. This formula is the basis for a relationship between the permittivity and permeability tensors needed to ahieve ray propagation along presribed paths using a urvilinear oordinate transformation [48]. Differential forms have impated other related topis suh as inverse sattering [49] and network theory [50]. In this paper, we review the notation, graphial representations, and key results that have arisen from the use of differential forms notation with Maxwell s equations and eletromagneti theory. We provide a basi introdution whih largely follows the presentation in some of the pedagogially oriented papers and books ited above, and disuss some of the insights that differential forms offer relative to

3 Progress In Eletromagnetis Researh, Vol. 148, key onepts from eletromagneti field theory. Finally, advaned topis inluding Green s funtions, potentials, and aperiodi spherial waves are treated. 2. DIFFERENTIAL FORMS The salar and vetor fields used in eletromagneti theory may be represented by exterior differential forms. Differential forms are an extension of the vetor onept. The use of differential forms does not neessarily replae vetor analysis. The differential form representation supplies additional physial insight in addition to the onventional vetor piture, and one one is familiar with the notation, it is easy to translate bak and forth between vetors and differential forms. One way to understand the differene between vetor fields and differential forms is to ontrast the two possible interpretations of the eletri field as a fore on a test harge (fore piture, represented by vetors) or the hange in energy experiened by the test harge as it moves through the field (energy piture, represented by differential forms). The two interpretations are not in ontradition, but omplement eah other in jointly providing a deep understanding of a fundamental physial property of the universe. The energy W 21 required to move a partile with eletri harge q in an eletri field with omponents E x, E y, and E z, from point r 1 to r 2 is obtained by the integral r2 W 21 (t) = q Ē dl (2) r 1 Figure 1(a) shows the path of integration for the definition of the voltage v 21 from node 2 to node 1. The vetor path integral sums up the projetion of the field vetor on the vetorial path element. The ontribution of the integrand over eah small length of the integration path is proportional to the produt of the magnitudes of field vetor with the infinitesimal path element and the osine of the angle enlosed between them. Instead of using a vetor, the eletri field an be represented by a linear ombination of differentials, aording to E = E x (x, y, z, t)dx + E y (x, y, z, t)dy + E z (x, y, z, t)dz (3) The objet on the right hand side is known as a one-form, and has three independent omponents, similar to a vetor. The key differene is that the unit vetors ˆx, ŷ, and ẑ are replaed by differentials dx, dy, and dz. The energy an be expressed in terms of the differential form as W 21 (t) = q E (4) r 1 whih is simpler than the vetor line integral, as there is no dot produt or differential line element. In a sense, the dot produt in (2) onverts the vetor into a differential that an be integrated. The integrand in (2) is already given in terms of differentials, so the dot produt is not needed. In this sense, the differential form representation is more natural than the vetor representation. In the differential forms notation, the eletri field is represented by a one-form. A one-form is a linear ombination of differentials of eah oordinate. An arbitrary one-form an be written r2 A = A 1 dx + A 2 dy + A 3 dz. (5) 2 E dx v 21 i(t) H (a) (b) Figure 1. Path of integration, (a) for the definition of the voltage and (b) for the definition of the urrent.

4 86 Warnik and Russer The three quantities A 1, A 2, and A 3 are the omponents of the one-form. Two one-forms A and B an be added, so that A + B = (A 1 + B 1 )dx + (A 2 + B 2 )dy + (A 3 + B 3 )dz. (6) One-forms an be integrated over paths. As shown in the introdution, we graphially represent a one-form as surfaes. The one-form dx has surfaes perpendiular to the x-axis spaed a unit distane apart. These surfaes are infinite in the y and z diretions. The integral of dx over a path from the point (0, 0, 0) to (4, 0, 0) is 4 0 dx = 4. This mathes the graphial representation in Figure 2(a), sine the path shown in the figure rosses four surfaes. If the path were not of integer length, we would have to imagine frational surfaes in between the unit spaed surfaes. A path from (0, 0, 0) to (0.25, 0, 0), for example, rosses 0.25 surfaes. We an also think of dx as a one-form in the plane. In this ase, the piture beomes a series of lines perpendiular to the x-axis spaed a unit distane apart, as shown in Figure 2(b). Graphially, integrals in the plane are similar to integrals in three dimensions: the value of a path integral is the number of lines piered by the path. In order to graphially integrate a one-form properly, we also have to think of the surfaes as having an orientation. The integral of the one-form dx over a path from (0, 0, 0) to (4, 0, 0) is 4. Thus, when we ount surfaes piered by a path, we have to ompare the sign of the one-form with the diretion of the path in order to determine whether the surfae ontributes positively or negatively. The orientation of surfaes an be indiated using an arrowhead on eah surfae, but sine the orientation is usually lear from ontext, to redue lutter we do not indiate it in figures. A more ompliated one-form, suh as 3dx + 5dy, has surfaes that are oblique to the oordinate axes. This one-form is shown in Figure 3. The greater the magnitude of the omponents of a one-form, the loser the surfaes are spaed. For one-forms with omponents that are not onstant, these surfaes an be urved. The surfaes an also originate along a line or urve and extend away to infinity, or the surfaes may be finite. In this ase, the integral over a path is still the number of surfaes or frational surfaes piered by the path. There are one-forms that annot be drawn as surfaes in three dimensions, but in most pratial situations, the surfae piture an be used. A one-form represents a quantity whih is integrated over a path, whereas vetor represents a quantity with a magnitude and diretion, suh as displaement or veloity. Despite this differene, both types of quantities have three independent omponents, and an be used interhangeably in desribing eletromagneti field quantities. Mathematially, vetors and differential forms are losely related. In Eulidean oordinates, we an make a orrespondene between vetors and forms. The one-form A and the vetor A are equivalent if they have the same omponents: A 1 dx + A 2 dy + A 3 dz A 1ˆx + A 2 ŷ + A 3 ẑ. (7) z y y x (a) (b) y x x Figure 2. (a) The one-form dx integrated over a path from the point (0, 0, 0) to (4, 0, 0). (b) The one-form dx in the plane. Figure 3. The one-form 3dx + 5dy.

5 ε Progress In Eletromagnetis Researh, Vol. 148, We say that the one-form A and the vetor A are dual. Sine it is easy to onvert between the differential form and vetor representations, one an hoose the quantity whih best suits a partiular problem. We will see in the next setion that in oordinate systems other then Eulidean, the duality relationship between forms and vetors hanges. The ommon physial interpretation of the eletri field is related to the fore on a point-like unit harge. This fore piture yields in a natural way to the vetor representation and to the visualization of the eletri field via field lines. Another viewpoint is to onsider the energy of a harge moved through the field. We an visualize the field via the hange of the energy of a test harge moved through the field. This energy piture is more related to differential forms. Figure 4(a) shows the representation of the field via the surfaes of onstant test harge energy or onstant eletri potential. For an eletrostati field the surfaes assoiated with the one-form E are equipotentials. The voltage between two points 2 and 1 is given by v 21 = 2 1 E. (8) For a stati eletri field, the surfaes of the one-form never meet or end. In general the surfaes of a oneform also may end or meet eah other, whih orresponds to fields produed by soures or time-varying fields. The magneti field intensity an be represented also by a one-form aording to H = H x (x, y, z, t)dx + H y (x, y, z, t)dy + H z (x, y, z, t)dz (9) The dimension of the differential form E is V and H has the dimension A. The oeffiients E x, E y, and E z have units of V/m, similar to the eletri field intensity vetor. The differentials dx, dy, and dz have units of length, whih ombines with the oeffiients so that the one-form E has units of V. Mathematially, the differential forms E and H express the hanges of the eletri and magneti potentials over an infinitesimal path element. Figure 5 shows a situation we enounter in time variable fields. In the enter of the struture the field intensity is higher than at its edges. New one-form surfaes appear as we move loser to the enter ε E D We e (a) (b) () E Figure 4. Geometri representation of (a) a oneform, (b) a two-form, () a three-form. The oneform is represented by surfaes, the two-form as tubes of flux or flow, and the three-form as ubes representing volumetri density. Figure 5. One-form with ending surfaes representing a nononservative field. H J Figure 6. A graphial representation of the physial behavior of magneti field and eletri urrent: tubes of J produe surfaes of H.

6 88 Warnik and Russer of the struture. In this ase the integral (8) will depend on the path from x 1 to x 2 and we annot assign a salar potential to the field. This type of field is sometimes referred to as nononservative. With the vetor piture, it an be hallenging to determine if a partiular field is nononservative, but with the one-form surfae representation, it is visually obvious. The magneti field produed by a line urrent behaves similarly, as new one-form surfaes of H are reated by the presene of an eletri urrent. one-form surfaes radiate outward from the line urrent. Figure 1(b) shows the path of integration for the definition of the urrent i. The integration path passes through a number of one-form surfaes equal to the amount of urrent flowing through the path. The differential forms piture of the line urrent and assoiated magneti field is shown in Figure 6. The surfaes of H emanate from the tubes of eletri urrent J. With the vetor piture, the vetor field orresponding to the magneti field intensity appear to rotate around the line urrent everywhere in spae, but the url of the vetor field is atually zero everywhere exept at the loation of the urrent. The differential forms piture makes it lear that the surfaes of the magneti field intensity one-form are reated by the tubes of urrent Integrating One-Forms over Paths The laws of eletromagnetis are expressed in terms of integrals of fields represented by differential forms. To apply these laws, we must be able to ompute the values of integrals of differential forms. Sine one-forms are by definition mathematial quantities whih are integrated over paths, the proess of evaluating an integral of a one-form is quite natural. The key idea is that we an replae the oordinates x, y, and z (or u, v, w in urvilinear oordinates) with a parametri equation for the integration path. A parametri representation of an integration path has the form [f(t), g(t), h(t)], so that the funtions f, g, and h give the oordinates of a point on the path for eah value of the parameter t. We replae the oordinates in a one-form by these funtions, and then the integral an be evaluated. For a differential, when the oordinate is replaed by a funtion defining the path, we then take the derivative by t to produe a new differential in the variable t. For example, dx beomes df = f (t)dt, where the prime denotes the derivative of the funtion f(t) by t. The differential form now has a single differential, dt, and the integral an be performed using standard rules of alulus. Two-forms an be integrated similarly, exept that the integration surfae requires two parameters, and the final integrand over the parametri surfae is a double integral Two-Forms, Three-Forms, and the Exterior Produt To introdue the onept of higher order differential forms, let us onsider the urrent i flowing in x-diretion through the surfae A in Figure 7(a). To ompute the urrent we have to integrate the x-omponent J x of the urrent density over the surfae A in the yz-plane i = J x dydz. (10) A If we integrate a urrent density over an area we have to onsider the orientation of the area. If in Figure 7(a) the urrent density J x is positive, the urrent i also will be positive. Inverting the diretion of J x will yield a negative urrent. This inversion may be performed by mirroring the oordinates with respet to the yz-plane. How do we know whether a surfae integral is positive or negative? The answer is: we have to define a positive orientation. A positive oriented or right-handed Cartesian oordinate system is speified as follows: if we are looking in z-diretion on the xy-plane the x-axis may be rotated lokwise by 90 into the y-axis. In Figure 7(a) the vetor omponent J x is pointing in positive orientation. In Figure 7(b) the oordinate system as well as the vetor field were rotated by 180 around the z-axis. Physially nothing has hanged. In the left figure, the vetor pointing towards the observer is positive, whereas in the right figure the vetor pointing away from the observer is positive. Exterior differential forms allow one to represent the orientation of a oordinate system in a rigorous way. We introdue the exterior produt or wedge produt dy dz with the property dy dz = dz dy (11)

7 Progress In Eletromagnetis Researh, Vol. 148, z z A A J x J x y y x x (a) (b) Figure 7. The orientation of an area. An exterior differential form is the exterior produt of differential forms. Exterior differential forms onsisting wedge produts of two differentials or sums of suh produts are alled two-forms. We may deide either dy dz = dydz or dy dz = dydz. Deiding that dy dz = dy dz (12) assigns to dy dz the positive orientation and to dz dy the negative orientation. The integral (10) an now be written in the orientation-independent form i = J x dy dz (13) A Physially, a two-form is a quantity whih is integrated over a two-dimensional surfae. A quantity representing flow of a fluid, for example, has units of flow rate per area, and would be integrated over a surfae to find the total flow rate through the surfae. Similarly, the integral of eletri flux density over a surfae is the total flux through the surfae, whih has units of harge. A general two-form C is written as C = C 1 dy dz + C 2 dz dx + C 3 dx dy. (14) For onveniene, we usually use the antisymmetry of the exterior produt to put the differentials of two-forms into right yli order, as in (14). We an use the exterior produt to reate differential forms of higher order as well. An exterior differential form of order p is alled a p-form and has p differentials in eah term. In n-dimensional spae the order of a differential form may assume values 0,..., n. In differential form notation a lear distintion between salars, pseudosalars, polar vetors and axial vetors is made. Salars are represented by zero-forms, pseudosalars by three-forms, polar vetors by one-forms and axial vetors by two-forms. For a p-form U and a q-form V the ommutation relation for the exterior produt is U V = ( 1) p+q+1 V U (15) Figure 8 shows the fundamental 1-, 2- and three-forms in Cartesian oordinates. dx dy dz dx dy dz (a) (b) () Figure 8. The fundamental (a) one-form, (b) two-form, () three-form in Cartesian oordinates.

8 90 Warnik and Russer 2.3. Current Density Two-Form We an desribe the flow of urrent in a ondutor by a urrent density vetor field J(x) = [J x (x), J y (x), J z (x)] T. The urrent flows along the urrent density field lines going through the boundary A of the area A as shown in Figure 9. In the differential forms representation, the urrent is depited as tubes of flow. Figure 4(b) shows the tube representation of a two-form. The two-form is visualized as a bundle of tubes arrying the urrent. Eah tube arries a unit amount of urrent, and the urrent density is inversely proportional to the ross-setional area of the tubes. Figure 8(b) shows the tube representations of the fundamental two-form dy dz. The tubes are made up of the one-form surfaes of dy and dz. A A J Figure 9. Current flow. If the surfae S is an arbitrarily oriented urved surfae in three-dimensional spae and the urrent density vetor has the x-, y- and z-omponents J x, J y and J z, the total urrent flowing through S is i = J x dy dz + J y dz dx + J z dx dy. (16) S The first term of the integrand onerns the integration of the x-omponent of the urrent density over the projetion of the surfae S on the yz-plane and so forth. Let us introdue the urrent density two-form J J = J x dy dz + J y dz dx + J z dx dy (17) The total urrent i flowing through S may be expressed in a ompat notation as the integral of the differential form J i = J (18) The dimension of the urrent density differential form J is A. The eletri flux density two-form D has the dimension As, and the magneti flux density two-form B has the dimension Vs. These differential forms represent the urrent or the flux through an infinitesimal area element Charge Density Three-Form The eletri harge q is given by the volume integral of the eletri harge density ρ. For the eletri harge density we introdue the eletri harge density three-form S Q = ρdx dy dz (19) Figure 4() shows the graphi visualization of a three-form by subdividing the volume into ells. The ell volume is inversely proportional to the harge density. The harge density form Q with the dimension As represents the harge in an infinitesimal volume element. We obtain the total harge q by performing the volume integral over the three-form Q: q = Q (20) V

9 Progress In Eletromagnetis Researh, Vol. 148, Eletromagneti Fields and Soures Tables 1 and 2 summarize the p-forms that desribe key eletromagneti field and soure quantities. The eletri and magneti fields are represented by both a one-form and a two-form. The one-form and two-form representations apture different physial properties of the field. The one-form represents the energy piture and shows the hange in potential energy as a test harge moves aross the one-form surfaes. The two-form shows the flux of the field that extends from positive harges to negative harges. Table 1. Differential forms that represent eletromagneti fields and soures with units and the orresponding vetor quantities. Quantity Form Type Units Vetor/Salar Eletri Field Intensity E one-form V E Magneti Field Intensity H one-form A H Eletri Flux Density D two-form C D Magneti Flux Density B two-form Wb B Eletri Current Density J two-form A J Eletri Charge Density Q three-form C ρ Table 2. Differential forms of eletromagnetism expanded in omponents. E = E x dx + E y dy + E z dz H = H x dx + H y dy + H z dz D = D x dy dz + D y dz dx + D z dx dy B = B x dy dz + B y dz dx + B z dx dy J = J x dy dz + J y dz dx + J z dx dy S = S x dy dz + S y dz dx + S z dx dy Q = ρ dx dy dz 2.6. Hodge Star Operator and the Constitutive Relations Sine the eletri field intensity E and the eletri flux density D represent the same field, it is lear that there must be some relationship between E and D. This relationship is expressed using the Hodge star operator. Simply stated, the star operator works by taking a differential form and onverting it to a new form with the missing differentials. For one-forms and two-forms, dx = dydz, dy = dzdx, dz = dxdy, dydz = dx dzdx = dy dxdy = dz The one-form A and the two-form A are both dual to the same vetor. The star operator applied twie is the identity operator, so that A = A for any form A. The star operator has a simple graphial interpretation. Considering the fundamental one-form dx, the surfaes of dx are perpendiular to the x diretion. Applying the star operator gives dydz, whih has tubes in the x diretion, so that the surfaes of dx are perpendiular to the tubes of dx. In general, the star operator ating on one-forms and two-forms onverts surfaes into perpendiular tubes and tubes into perpendiular surfaes. This is illustrated in Figure 10. The relation between the eletri field intensity one-form E and the eletri flux density D depends on the material properties of the medium and the metri properties of spae. In an isotropi medium, the eletri field intensity vetor and the flux density vetor are proportional, and it is not always lear why the two different quantities are needed for the eletri field. With differential forms notation,

10 92 Warnik and Russer Figure 10. The star operator onverts one-form surfaes into perpendiular two-form tubes. field intensity and flux density are represented by different mathematial objets. The mathematial relationship between E and D is known as a onstitutive relation. The onstitutive relation is written using the Hodge star operator as D = ɛ E (21) where ɛ is the permittivity of a medium. The permittivity ɛ aounts for the material properties whereas the Hodge operator aounts for the metri properties of the spae or the hosen oordinate system, respetively. This equation shows that surfaes of E yield perpendiular tubes of D (see Figure 10). The onstant sales the sizes of the tubes without hanging their diretion. The magneti field onstitutive relation is B = µ H. (22) where µ is the permeability of the medium. The differential forms notation helps to eluidate why two different representations are used for one physial field. With vetor notation, flux and field intensity are both vetors, but with differential forms, they beome one-forms and two-forms, with different geometrial representations and physial interpretations. 3. MAXWELL S EQUATIONS IN INTEGRAL FORM The integral form of Maxwell s equations is given by: H = d S dt D + J, Ampère s Law (23) S B, Faraday s Law (24) S E = d S dt S B = 0, Magneti Flux Continuity (25) V D = Q. Gauss Law (26) V V Ampère s law relates urrent to magneti field. It states that the sum of ondution and displaement urrents through an area S equals the magneti tension around the boundary S. Faraday s law relates the magneti flux to the eletri field. It says that the time derivative of the magneti flux through an area S equals the eletri tension around the boundary S. Both equations are tied together via the onstitutive equations relating flux densities to field intensities Point Charge As a simple example of the appliation of Maxwell s laws in integral form, we onsider the eletri flux produed by a point harge. The tubes of flux from a point harge Q extend out radially from the

11 Progress In Eletromagnetis Researh, Vol. 148, Figure 11. Eletri flux density due to a point harge. Tubes of D extend out radially from the harge. harge (Figure 11). Thus, the flux density two-form D has to be a multiple of the differentials dθdφ in the spherial oordinate system. The tubes of dθdφ are denser near the poles θ = 0 and θ = π than at the equator. We need to inlude the orretion fators r 2 sin θ to have tubes with the same density everywhere in spae. Thus, D has the form D = D 0 r 2 sin θdθdφ (27) where D 0 is a onstant. To apply Gauss law, we will hoose the surfae S to be a sphere around the harge. The right-hand side is Q = Q (28) where V is the volume inside the sphere. The left-hand side is 2π π D = D 0 r 2 sin θdθdφ = 4πr 2 D 0. S 0 0 V By Gauss s law, we know that 4πr 2 D 0 = Q. Solving for D 0 and substituting into (27), we obtain D = Q sin θdθdφ (29) 4π for the eletri flux density due to the point harge. Sine 4π is the total amount of solid angle for a sphere and sin θdθdφ is the differential element of solid angle, this expression implies that the amount of flux per solid angle is the same at any distane from the harge. 4. THE EXTERIOR DERIVATIVE In this setion, we define the exterior derivative operator, whih will allow us to express Maxwell s laws as differential equations. This operator has the symbol d, and ats on a p-form to produe a new form with degree p + 1. This (p + 1)-form haraterizes the spatial variation of the p-form. The exterior derivative operator an be written as ( d = x dx + y dy + ) z dz (30) The d operator is similar to a one-form, exept that the oeffiients are partial derivative operators instead of funtions. When this operator is applied to a differential form, the derivatives at on the oeffiients of the form, and the differentials ombine with those of the form aording to the properties of the exterior produt. The sum and produt rules for exterior differentiation are: d (U + V) = du + dv, d (U V) = du V + ( 1) (deg U) U dv. (31a) (31b)

12 94 Warnik and Russer The exterior derivatives of p-forms are zero-form: df(x) = f f f xdx + y dy + z dz, one-form: du(x) = two-form: dv(x) = ( Uz y ( Vx x Uy z + Vy y ) dy dz + ( U x z ) + V z z dx dy dz, ) ( Uz Uy x dz dx + x ) Ux y dx dy, three-form: dq(x) = 0. Computing exterior derivatives is straightforward. One takes the partial derivative of a differential form by x and adds the differential dx from the left, repeats for y and z, and adds the three results. This proess is very similar to impliit differentiation, exept that one must ombine new differentials with existing differentials using the exterior produt. When taking the exterior derivative of a one-form or two-form, some terms may drop out due to repeated differentials. Table 3 shows the orrespondene between several vetor derivative and exterior derivative operations. Table 3. Differential operators. Vetor Differential Operator Exterior Differential Operator grad f df url A da div B db url grad f = 0 d df = 0 div url A = 0 d da = 0 div grad f d df or d df url url A d da or d da 5. POINCARÉ S LEMMA A form V for whih dv = 0 is said to be losed, and a form V for whih V = du is said to be exat. For differential forms the statement V = du implies dv = 0. The relation dd U = 0 (32) may be verified easily. In onventional vetor notation this orresponds to url grad = 0 and div url = 0. All exat forms are losed. However it may also be shown that all losed forms are exat. Poinaré s lemma states that dv = 0 V = du. (33) If a differential form has zero exterior derivative, then it is itself the exterior derivative of another form. 6. STOKES THEOREM In (23) and (24) line integrals over the boundary of the surfae A are related to surfae integrals over A. Figure 12(a) shows the relation between the orientation of the area A and the boundary A. The line integral over the losed ontour A is alled irulation. In (25) and (26) the surfae integrals are performed over the boundary V of the volume V. Figure 12(b) shows the orientation of the boundary surfae V. The Stokes theorem relates the integration of a p-form U over the losed p-dimensional boundary V of a (p + 1)-dimensional volume V to the volume integral of du over V via U = d U. (34) V V

13 Progress In Eletromagnetis Researh, Vol. 148, dx dy dz V A A V (a) dx dy (b) Figure 12. (a) Area A with boundary A and (b) volume V with boundary V. (a) (b) (a) (b) Figure 13. Stokes theorem for U a one-form. (a) The loop V pieres three of the surfaes of U. (b) Three tubes of du pass through a surfaes V bounded by the loop V. Figure 14. Stokes theorem for U a two-form. (a) Four tubes of U pass through the losed, retangular surfae V. (b) Four boxes of the three-form du lie inside the surfae. This summarizes the Stokes theorem and the Gauss theorem of onventional vetor notation. The property of Stokes theorem to relate the integration of a p-form over a losed boundary of a volume to the integration of the exterior derivative of the p-form over the volume enlosed by that boundary yields an interesting geometri interpretation of Poinaré s lemma: sine a boundary has no boundary the iterated appliation of the exterior derivative yields zero value. If U is a zero-form, then Stokes theorem states that b a du = f(b) f(a). This is the fundamental theorem of alulus. If U is a one-form, then V has to be a losed path. V is a surfae that has the path as its boundary. Graphially, Stokes theorem says that the number of surfaes of U piered by the path is equal to the number of tubes of the two-form du that pass through the path (Figure 13). If U is a two-form, then V is a losed surfae and V is the volume inside it. Stokes theorem requires that the number of tubes of U that ross the surfae is equal to the number of boxes of du inside the surfae, as shown in Figure CURVILINEAR COORDINATES In exterior alulus the field equations may be formulated without referene to a speifi oordinate system. Depending on the problem the hoie of a speifi oordinate system may simplify the problem solution onsiderably. We introdue an orthogonal urvilinear oordinate system u = u(x, y, z), v = v(x, y, z), w = w(x, y, z). (35) The oordinate urves are obtained by setting two of the three oordinates u, v, w onstant. Coordinate surfaes are defined by setting one of the three oordinates onstant. In an orthogonal oordinate system in any point (exept singular points) of the spae the three oordinate urves are orthogonal. The same

14 96 Warnik and Russer holds for the three oordinate surfaes going through any point. The differentials dx, dy, dz by the differentials du, dv, dw are related to dx = x x x du + dv + dw, (36) u v w dy = y y y du + dv + dw, (37) u v w dz = z z z du + dv + dw. (38) u v w The rules for transformation of the Cartesian basis two-forms dx dy, dy dz, dz dx and the Cartesian basis three-form dx dy dz follow diretly from the above equations by applying the rules of the exterior produt. Using the metri oeffiients g 1, g 2 and g 3 g1 2 = x u x u, we introdue the unit one-forms g2 2 = x v x v, g2 3 = x w x w (39) s 1 = g 1 du, s 2 = g 2 dv, s 3 = g 3 dw. (40) The integral of s 1 = g 1 du along any path with v and w onstant yields the length of the path. For the urvilinear unit differentials the Hodge star operator is f = fs 1 s 2 s 3, (A u s 1 + A v s 2 + A w s 3 ) = A u s 2 s 3 + A v s 3 s 1 + A w s 1 s 2, (A u s 2 s 3 + A v s 3 s 1 + A w s 1 s 2 ) = A u s 1 + A v s 2 + A w s 3, (fs 1 s 2 s 3 ) = f. The appliation of the Hodge operator fully aounts for the metri properties of the used oordinate system. In ontrast to this the exterior derivative is independent of the metri. This is a substantial advantage of exterior alulus Cylindrial Coordinates In the ylindrial oordinate system, a point in spae is speified by the radial distane of its x, y oordinates ρ = x 2 + y 2, angle from the +x axis in the x-y plane φ, and height in the z diretion. The differentials of the ylindrial oordinate system are dρ, dφ and dz. To onvert forms into unit vetors, the angular differential dφ must be made into a unit differential ρdφ. One-forms orrespond to vetors by the rules dρ ˆρ ρdφ ˆφ dz ẑ Figure 15 shows the pitures of the differentials of the ylindrial oordinate system. The two-forms an be obtained by superimposing these surfaes. Tubes of dz dρ, for example, are square donut-shaped and point in the φ diretion. In urvilinear oordinates, differentials must be onverted to unit differentials before the star operator is applied. The star operator in ylindrial oordinates ats as follows: dρ = ρdφ dz ρdφ = dz dρ dz = dρ ρdφ so that ρ must be inluded with dφ before the star operator an be applied to make it a two-form with unit differentials. Also, 1 = ρ dρdφdz. As with the retangular oordinate system, = 1, so that this same table an be used to onvert two-forms to one-forms. (41)

15 Progress In Eletromagnetis Researh, Vol. 148, z z z y y dρ x (a) z x (b) ρ dϕ dz y y x () x Figure 15. Surfaes of dρ, dφ saled by 3/π and dz. Figure 16. Unit differentials in ylindrial oordinates represented as faes of a differential volume Spherial Coordinates In the spherial oordinate system, a point in spae is speified by the radial distane from the origin r = x 2 + y 2 + z 2, angle from the +x axis in the x-y plane φ, and angle from the z axis θ. The differentials of the spherial oordinate system are dr, dθ and dφ. To onvert forms into unit vetors, the angular differentials must be made into unit differentials rdθ and r sin θdφ. One-forms orrespond to vetors by the rules dr ˆr rdθ ˆθ r sin θdφ ˆφ Figure 17 shows the pitures of the differentials of the spherial oordinate system. As in the ylindrial oordinate system, differentials of the spherial oordinate system must be onverted to length elements before the star operator is applied. The star operator ats on one-forms and two-forms as follows: dr = rdθ r sin θdφ rdθ = r sin θdφ dr r sin θdφ = dr rdθ Again, = 1, so that this table an be used to onvert two-forms to one-forms. The star operator applied to one is r 2 sin θdrdθdφ Exterior Derivative in Curvilinear Coordinates Regardless of the partiular oordinate system, the form of the exterior derivative operator remains the same. If the oordinates are (u, v, w), the exterior derivative operator is ( d = u du + v d + ) w dw (42) In ylindrial oordinates, d = ( ρ dρ + φ dφ + ) z dz (43)

16 θ 98 Warnik and Russer z z z y y rsin dϕ x (a) z x (b) rdθ dr y y x () x Figure 17. Surfaes of dr, dθ saled by 10/π and dφ saled by 3/π. Figure 18. Unit differentials in spherial oordinates represented as faes of a differential volume. whih is the same as for retangular oordinates but with the oordinates ρ, φ, z in the plae of x, y, z. Note that the fator ρ assoiated with dφ must be present when onverting to vetors or applying the star operator, but is not found in the exterior derivative operator. The exterior derivative in spherial oordinates is ( d = r dr + θ dθ + ) φ dφ. (44) From these expressions, we see that omputing exterior derivatives in a urvilinear oordinate system is no different from omputing in retangular oordinates. 8. MAXWELL S EQUATIONS IN LOCAL FORM Applying Stokes theorem to the integral form of Maxwell s Equations (23) to (26) we obtain the differential representation of Maxwell s equations: d H = D + J, Ampère s Law (45) t d E = B, t Faraday s Law (46) d B = 0, Magneti Flux Continuity (47) d D = Q. Gauss Law (48) Figure 19 shows the graphial representation of Maxwell s equations after Deshamps [51]. We have two groups of equations, namely Faraday s law and magneti flux ontinuity on the left-hand side of the diagram and Ampére s law and Gauss law on the right-hand side. Within eah group exterior differentiation is metri-independent and inreases the order of the respetive p-forms. Both groups are related via the onstitutive relations depending on the material properties and the oordinate metris and either inrementing or derementing the order of the respetive p-forms.

17 ε Progress In Eletromagnetis Researh, Vol. 148, Faraday s Law Magneti Flux Continuity Ampere's Law Gauss' Law ' d Constitutive Relations zero-form / d t ε µ d d / t / t 0 d d d one-form two-form 0 t 0 three-form Figure 19. Graphial representation of Maxwells equations. 9. BOUNDARY CONDITIONS If a magneti field hanges abruptly along some boundary surfae, Maxwell s laws require that an eletri urrent flow along the boundary to aount for the step in field intensity. Similarly, Maxwell s laws restrit the possible disontinuity in the eletri field at a boundary. In this setion, we derive expressions for these boundary onditions on E, H, D, and B Field Intensity In this setion, we derive boundary onditions for the eletri and magneti field intensity one-forms E and H. As in Figure 20, we denote the magneti field on one side of a boundary as H 1, and the field on the other side as H 2. If we hoose an Amperian ontour with one side just above and the other just below the boundary, as shown in Figure 20, the left hand side of Ampere s law beomes H = (H 1 H 2 ) (49) C P in the limit as the width of the ontour goes to zero and the sides of the ontour meet eah other along a ommon path P on the boundary. The right hand side is equal to the surfae urrent flowing aross the path P, so that (H 1 H 2 ) = J s (50) P P H 1 H 2 (a) (b) Figure 20. A disontinuity in the magneti field above and below a boundary. The surfae urrent flowing on the boundary an be found using an Amperian ontour with infinitesimal width. Figure 21. (a) The one-form H 1 H 2. (b) The one-form J s, represented by lines on the boundary. Current flows along the lines.

18 100 Warnik and Russer where JJ s is a one-form representing the surfae urrent density on the boundary. Sine (50) holds for any path P on the boundary, the integrands must be equal on the boundary. We thus have that J s = (H 1 H 2 ) B (51) where the right hand side is the restrition of the magneti field disontinuity to the boundary. The one-form J s is represented graphially by the lines along whih the one-form H 1 H 2 intersets the boundary, as shown in Figure 21. Current flows along these lines. If the surfaes of H 1 H 2 are parallel to the boundary, then the surfaes do not interset, and the restrition is zero. Thus, (51) represents the tangential omponent of the magneti field disontinuity. The diretion of urrent flow along these lines an be obtained using the right hand rule: if the right hand is on the boundary and the fingers point in the diretion of H 1 H 2, then the thumb points in the diretion of urrent flow. In order to ompute the restrition mathematially, we employ an expression of the form z = f(x, y) to represent the boundary, and replae all ourenes of the variable z in H 1 H 2 with the funtion f(x, y), so that J s = [H 1 (x, y, z) H 2 (x, y, z)] z=f(x,y) = [H 1x (x, y, f) H 2x (x, y, f)]dx + [H 1y (x, y, f) H 2y (x, y, f)]dy + [H 1z (x, y, f) H 2z (x, y, f)]df [ = H 1x H 2x + f ] [ x (H 1z H 2z ) dx + H 1y H 2y + f ] y (H 1z H 2z ) dy If part of the boundary is parallel to the x y plane, then the boundary must be expressed as x = g(y, z) or y = h(x, z). In a similar manner, we an show that the eletri field satisfies the boundary ondition (E 1 E 2 ) B = 0 (52) This ondition requires that the tangential omponent of the eletri field above and below a boundary must be equal at the boundary Flux Density From Gauss s law, it an be shown that the eletri flux density satisfies the boundary ondition (D 1 D 2 ) B = Q s (53) where Q s is a two-form representing the density of eletri surfae harge on the boundary. This twoform is represented graphially as boxes whih are the intersetion of the tubes of D 1 D 2 with the boundary, as in Figure 22. If the tubes of the magneti flux disontinuity are parallel to the boundary, then the tubes do not interset and the restrition is zero. The left-hand side of (53) is the omponent of the jump in flux whih is normal to the boundary. The magneti flux density satisfies the boundary ondition (B 1 B 2 ) B = 0 (54) so that the normal omponents of the magneti flux above and below a boundary must be equal. (a) (b) Figure 22. (a) The two-form D 1 D 2. (b) The two-form Q s, represented by boxes on the boundary.

19 ε Progress In Eletromagnetis Researh, Vol. 148, We ollet all of the boundary onditions for referene: (E 1 E 2 ) B = 0 (H 1 H 2 ) B = J s (D 1 D 2 ) B = Q s (B 1 B 2 ) B = 0 The first two involve the tangential omponent of the field intensity, and the seond pair involve the normal omponent of flux density. These onditions are simply onvenient restatements of Maxwell s laws for fields at a boundary. 10. ENERGY AND POWER The eletri and the magneti energy densities are represented by the three-forms W e = 1 2 E D = 1 2 (E xd x + E y D y + E z D z ) dx dy dz, W m = 1 2 H B = 1 2 (H xb x + H y B y + H z B z ) dx dy dz. (55a) (55b) Figure 23 visualizes the exterior produt of the field one-form E and the flux density two-form D. The resulting energy density three-forms W e and W m are visualized by the subdivision of the spae into ells as shown in Figure 23. Multiplying Ampère s law from the left with E and Faraday s law from the right with H, we obtain E dh = t D + J, (56) de = B H. (57) t This yields d (E H) = E t D H t B E J. (58) This equation an be brought into the form d (E H) = t The power loss density p L (x, t) with the orresponding differential form is given by ( 1 2 E D + 1 ) 2 H B E J. (59) P L = p L (x, t) dx dy dz (60) P L = E σ E. (61) Figure 23. The exterior produt of the field form E and the flux density form D.

20 ε 102 Warnik and Russer = ε Figure 24. The Poynting form S as the produt of the field forms E and H. Due to the impressed urrent density J 0, a power per unit of volume P 0 = E J 0 (62) is added to the eletromagneti field. Introduing the Poynting differential form S = E H (63) and inserting (55a), (55b), (61) and (62) into (59) yields the differential form of Poynting s theorem d S = t W e t W m P L + P 0. (64) Figure 24 visualizes the Poynting two-form as the exterior produt of the eletri and magneti field one-forms E and H. The potential planes of the eletri and magneti fields together form the tubes of the Poynting form. The distane of the eletri and magneti potential planes exhibit the dimensions V and A respetively. The ross setional areas of the flux tubes have the dimension VA. The power flows through these Poynting flux tubes. Integrating (64) over a volume V and transforming the integral over S into a surfae integral over the boundary V, we obtain the integral form of Poynting s Theorem S = P 0 d W e d W m P L. (65) V dt V dt V V V 11. TELLEGEN S THEOREM Figure 25 shows the segmentation of an eletromagneti struture into different regions R l separated by boundaries B lk. The regions R l may ontain any eletromagneti substruture. In a network analogy the two-dimensional manifold of all boundary surfaes B lk represents the onnetion iruit, whereas the subdomains V l are representing the iruit elements. Tellegen s theorem states fundamental relations between voltages and urrents in a network and is of onsiderable versatility and generality Figure 25. Segmentation of a losed struture.

21 Progress In Eletromagnetis Researh, Vol. 148, in network theory [52]. The field form of Tellegen s theorem may be derived diretly from Maxwell s equations [50, 53] and is given by E ( x, t ) H ( x, t ) = 0. (66) V The integration is performed over both sides of all boundary surfaes. Also the integration over finite volumes filled with ideal eletri or magneti ondutors gives no ontribution to these integrals. The prime and double prime denote the ase of a different hoie of soures and a different hoie of materials filling the subdomains. Also the time argument may be different in both ases. 12. THE ELECTROMAGNETIC POTENTIALS The Maxwell s Equations (45) (48) are a system of twelve oupled salar partial differential equations. The introdution of eletromagneti potentials allows a systemati solution of the Maxwell s equations [54 56]. We are distinguishing between salar potentials and vetor potentials. After solution of the wave equation for a potential, all field quantities may be derived from this potential. Due to (47), i.e., db = 0 the magneti flux density is free of divergene. Therefore B may be represented as the exterior derivative of an one-form A: B = da. (67) The orresponding vetor field A is alled the magneti vetor potential. Inserting (67) into the seond Maxwell s Equation (46) yields d (E + t ) A = 0. (68) Aording to Poinaré s lemma, the exterior derivative of the one-form inside the brakets vanishes, we may express this one-form as the exterior derivative of the salar potential Φ and obtain E = dφ A. (69) t The negative sign of Φ has been hosen due to the physial onvention in defining potentials. Whereas in eletrostatis the eletri field may be omputed from a salar potential Φ, in the ase of rapidly varying eletromagneti fields, we also need the vetor potential A. The potentials A and Φ are not defined in an unambiguous way. Adding the gradient of a salar funtion Ψ to the vetor potential A does not influene the magneti indution B. The eletri field E also remains unhanged, if A and Φ together are transformed in the following way: A 1 = A + dψ, (70) Φ 1 = Φ Ψ t. (71) This transformation is alled a gauge transformation. The one-form A may be defined in an unambiguous way, if we are presribing its exterior derivative. Inserting (67) and (69) into the first Maxwell s Equation (45) yields d da + µε 2 t 2 A + µσ ( t A + µd ε Φ ) t + σφ = µj 0. (72) Inserting (69) and (21) into (48) yields d dφ + d t A = 1 Q. (73) ε Sine we may hoose the exterior derivative of A arbitrarily, we an make use of this option in order to deouple the differential equations for A and Φ. We impose the so-alled Lorentz ondition given by d A + µ (ε t ) Φ + σφ = 0. (74)